Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, C

Average Error: 0.01% → 0%

Time: 4.1s

Precision: binary64

Cost: 6656

Math

\[x - y \cdot z \]

↓

\[\mathsf{fma}\left(-z, y, x\right) \]

FPCore

(FPCore (x y z) :precision binary64 (- x (* y z)))

↓

(FPCore (x y z) :precision binary64 (fma (- z) y x))

C

double code(double x, double y, double z) {
	return x - (y * z);
}

↓

double code(double x, double y, double z) {
	return fma(-z, y, x);
}

Julia

function code(x, y, z)
	return Float64(x - Float64(y * z))
end

↓

function code(x, y, z)
	return fma(Float64(-z), y, x)
end

Wolfram

code[x_, y_, z_] := N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[((-z) * y + x), $MachinePrecision]

Target

Original	0.01%
Target	0.02%
Herbie	0%

\[\frac{x + y \cdot z}{\frac{x + y \cdot z}{x - y \cdot z}} \]

Derivation

1. Initial program 0.01

   \[x - y \cdot z \]

2. Applied egg-rr 2.1

   \[\leadsto \color{blue}{{\left(\sqrt[3]{x - y \cdot z}\right)}^{2} \cdot \sqrt[3]{x - y \cdot z}} \]

3. Applied egg-rr 0

   \[\leadsto \color{blue}{y \cdot \left(-z\right) + \left(x + \mathsf{fma}\left(-z, y, y \cdot z\right)\right)} \]

4. Taylor expanded in y around 0 0.07

   \[\leadsto \color{blue}{\left(-2 \cdot z + z\right) \cdot y + x} \]

5. Simplified 0

   \[\leadsto \color{blue}{\mathsf{fma}\left(-z, y, x\right)} \]
   Proof

   [Start] 0.07	\[ \left(-2 \cdot z + z\right) \cdot y + x \]
   distribute-lft1-in [=>] 0.01	\[ \color{blue}{\left(\left(-2 + 1\right) \cdot z\right)} \cdot y + x \]
   metadata-eval [=>] 0.01	\[ \left(\color{blue}{-1} \cdot z\right) \cdot y + x \]
   fma-def [=>] 0	\[ \color{blue}{\mathsf{fma}\left(-1 \cdot z, y, x\right)} \]
   mul-1-neg [=>] 0	\[ \mathsf{fma}\left(\color{blue}{-z}, y, x\right) \]

6. Final simplification 0

   \[\leadsto \mathsf{fma}\left(-z, y, x\right) \]

Alternatives

Alternative 1
Error	29.27%
Cost	520

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-87}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+94}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.01%
Cost	320

\[x - z \cdot y \]

Alternative 3
Error	42.08%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023089 
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, C"
  :precision binary64

  :herbie-target
  (/ (+ x (* y z)) (/ (+ x (* y z)) (- x (* y z))))

  (- x (* y z)))